Metrically Stationary, Axially Symmetric, Isolated Systems in Quasi-Metric Gravity

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2 Quasi-metric gravity described succinctly

Quasi-metric relativity (QMR), including its current observational status, is described in detail elsewhere [2, 3, 4]. Here we give a very brief survey and include only the formulae needed for calculations. The basic idea which acts as a motivation for postulating the quasi-metric geometrical framework is that the cosmic expansion should be described as an inherent geometric property of quasi-metric space-time itself and not as a kinematical (in the general sense of the word) phenomenon subject to dynamical laws. That is, in QMR, the cosmic expansion is described as a general phenomenon that does not have a cause. This means that its description should not depend on the causal structure associated with any pseudo- Riemannian manifold. Such an idea is attractive since in this way, one should be able to avoid the in principle enormous multitude of possibilities, regarding cosmic genesis, initial conditions and evolution, present if space-time is modelled as a pseudo-Riemannian manifold. Therefore, one expects that any theory of gravity compatible with the quasi- metric framework is more predictive than any metric theory of gravity when it comes to cosmology. The geometric basis of the quasi-metric framework consists of a 5-dimensional differen- tiable product manifold R , where = R is a (globally hyperbolic) Lorentzian M× 1 M S× 2 space-time manifold, R and R are two copies of the real line and is a compact Rie- 1 2 S mannian 3-dimensional manifold (without boundaries). The global time function t is then introduced as a coordinate on R1. The product topology of implies that once t is M 0 0 given, there must exist a “preferred” ordinary time coordinate x on R2 such that x scales like ct. A coordinate system on with a global time coordinate of this type we M call a global time coordinate system (GTCS). Hence, expressed in a GTCS xµ (where { } 2 µ can take any value 0 3), x0 is interpreted as a global time coordinate on R and xj − 2 { } (where j can take any value 1 3) as spatial coordinates on . The class of GTCSs is a − S set of preferred coordinate systems inasmuch as the equations of QMR take special forms in a GTCS. The manifold R is equipped with two degenerate 5-dimensional metrics ¯g and M× 1 t gt. By definition the metric ¯gt represents a solution of field equations, and from ¯gt one can construct the “physical” metric gt which is used when comparing predictions to experiments. To reduce the 5-dimensional space-time R to a 4-dimensional space- M× 1 time we just slice the 4-dimensional sub-manifold determined by the equation x0 = ct N (using a GTCS) out of R . It is essential that there is no arbitrariness in this choice M× 1 of slicing. That is, the identification of x0 with ct must be unique since the two global time coordinates should be physically equivalent; the only reason to separate between them is that they are designed to parameterize fundamentally different physical phenomena. Note that is defined as a compact 3-dimensional manifold to avoid ambiguities in the S slicing of R . M× 1 Moreover, in , ¯g and g are interpreted as one-parameter metric families (this inter- N t t pretation is merely a matter of semantics). Thus by construction, is a 4-dimensional N space-time manifold equipped with two one-parameter families of Lorentzian 4-metrics parameterized by the global time function t. This is the general form of the quasi-metric space-time framework. We will call a quasi-metric space-time manifold. One reason N why cannot be represented by single Lorentzian manifolds is that the affine connection N compatible with any metric family is non-metric; see [2, 3] for more details. From the definition of quasi-metric space-time we see that it is constructed as consisting of two mutually orthogonal foliations: on the one hand space-time can be sliced up globally into a family of 3-dimensional space-like hypersurfaces (called the fundamental hypersurfaces (FHSs)) by the global time function t, on the other hand space-time can be foliated into a family of time-like curves everywhere orthogonal to the FHSs. These curves represent the world lines of a family of hypothetical observers called the fundamental observers (FOs), and the FHSs together with t represent a preferred notion of space and time. That is, the equations of any theory of gravity based on quasi-metric geometry should depend on quantities obtained from this preferred way of splitting up space-time into space and time.

The metric families ¯gt and gt may be decomposed into parts respectively normal to and intrinsic to the FHSs. The normal parts involve the unit normal vector ﬁeld families ¯n and n of the FHSs, with the property ¯g (¯n , ¯n )= g (n , n )= 1. The parts intrinsic t t t t t t t t −

3 to the FHSs are the spatial metric families h¯t and ht, respectively. We may then write

¯g = ¯g (¯n , ) ¯g (¯n , )+ h¯ , (1) t − t t · ⊗ t t · t and similarly for the decomposition of gt. Moreover, expressed in a GTCS, ¯nt may be written as (using the Einstein summation convention) ∂ ∂ t ∂ ¯n n¯µ = N¯ −1 0 N¯ k , (2) t≡ (t) ∂xµ t ∂x0 − t ∂xk where t0 is an arbitrary reference epoch, N¯t is the family of lapse functions of the FOs and t0 N¯ k are the components of the shift vector field family of the FOs in ( , ¯g ). (A similar t N t formula is valid for n .) Note that in the rest of this paper we will use the symbol ‘ ¯ ’ to t ⊥ mean a scalar product with ¯n . A useful quantity derived from N¯ is the 4-acceleration − t t field ¯a of the FOs in ( , ¯g ) which is a quantity intrinsic to the FHSs. Expressed in a F N t GTCS, ¯aF is defined by its components ¯ −2 Nt,j −1 k c a¯Fj , y¯t c a¯Fka¯F , (3) ≡ N¯t ≡ q where its norm is given by cy¯t and where a comma denotes partial derivation.

For reasons explained in [2, 3], the form of ¯gt is restricted such that it contains only one dynamical degree of freedom. That is, gravity is required to be essentially scalar in ( , ¯g ). Besides, the geometry of the FHSs is deﬁned to represent a gravitational scale N t as measured in atomic units. Therefore ¯gt takes an even more restricted form. In fact, expressed in a GTCS, the most general form allowed for the family ¯gt is represented by the family of line elements (this may be taken as a deﬁnition)

2 2 ¯ ¯ s ¯ 2 0 2 t ¯ i 0 t ¯ 2 i k dst = NsN Nt (dx ) +2 Nidx dx + 2 Nt Sikdx dx , (4) − t0 t0 h i i k 3 ¯ ¯ 2 ¯ k where Sikdx dx is the metric of S (with radius equal to ct0) and Ni Nt SikN . Also note t ≡ that, by formally setting t0 = 1 and replacing the metric of the 3-sphere with an Euclidean 3-metric in equation (4), we get a single metric which represents the correspondence with metric gravity. By definition this so-called metric approximation is possible whenever a correspondence with metric theory can be found by approximating tensor field families by single tensor fields not depending on t. j Field equations from which N¯t and N¯ can be determined are given as couplings between projections of the Ricci tensor family R¯ t and the active stress-energy tensor Tt. However, dimensional analysis yields that the gravitational coupling must be nonuniversal [2, 3]. That is, there must be two different (variable) gravitational coupling parameters

4 B S Gt and Gt for coupling of different types of matter sources to space-time geometry. To B (EM) be more specific, Gt couples to the active electromagnetic stress-energy tensor Tt S mat while Gt couples to the active stress-energy tensor Tt representing material sources [2, 3]. The field equations then read (using a GTCS)

−2 i −4 i ik 2R¯ ¯ ¯ = 2(c a¯ + c a¯ a¯ K¯ K¯ + £¯nt K¯ ) (t)⊥⊥ F|i Fi F − (t)ik (t) t B (EM) ˆ(EM)i S mat ˆmati = κ (T(t)⊥¯ ⊥¯ + T(t)i )+ κ (T(t)⊥¯ ⊥¯ + T(t)i ), (5)

i B (EM) S mat R¯ ¯ = K¯ K¯ , = κ T + κ T , (6) (t)j⊥ (t)j|i − t j (t)j⊥¯ (t)j⊥¯ ¯ where £¯nt denotes Lie derivation in the direction normal to the FHS and Kt is the extrinsic curvature tensor family (with trace K¯ ) of the FHSs. Moreover κB 8πGB/c4, t ≡ κS 8πGS/c4, a “hat” denotes an object projected into the FHSs and the symbol ‘ ’ ≡ B B S S | denotes spatial covariant derivation. The values G of Gt and G of Gt , respectively, are by convention chosen as those measured in (hypothetical) local gravitational experiments in an empty universe at epoch t0. Note that all quantities correspond to the metric family

¯gt.

It is useful to have an explicit expression for K¯ t, which may be calculated from equation (4). Using a GTCS we ﬁnd

¯ ¯ k t Nt,⊥¯ t0 −2 N ¯ K¯(t)ij = (N¯i|j + N¯j|i)+ c a¯Fk h(t)ij , (7) 2t0N¯t N¯t − t N¯t

N¯ i ¯ ¯ k t0 |i Nt,⊥¯ t0 −2 N K¯t = +3 c a¯Fk . (8) t N¯t N¯t − t N¯t It is also convenient to have explicit expressions for the curvature intrinsic to the FHSs. From equation (4) one easily calculates 1 H¯ = c−2 a¯k h¯ c−4a¯ a¯ c−2a¯ , (9) (t)ij F|k ¯ 2 2 (t)ij Fi Fj Fi|j − Nt t − − 6 P¯ = +2c−4a¯ a¯k 4c−2a¯k , (10) t 2 Fk F F|k (N¯tct) − where H¯ t is the Einstein tensor family intrinsic to the FHSs in ( , ¯gt). N ⋆ The coordinate expression for the covariant divergence of T , i.e., ¯ T , reads t ∇· t T ν T ν + c−1T 0 , (11) (t)µ∗¯ν≡ (t)µ;ν (t)µ∗¯t 5 where the symbol ‘¯’ denotes degenerate covariant derivation compatible with the family ∗ ¯gt and a semicolon denotes metric covariant derivation in component notation. Moreover we have that [2, 3] ¯ 0 2 1 Nt,t T(t)µ∗¯t = + T(t)⊥¯ µ, (12) −N¯t t N¯t and also that [2, 3] ¯ ¯ ν Nt,ν ν −2 ˆi Nt,⊥¯ T(t)µ;ν =2 T(t)µ =2c a¯FiT(t)µ 2 T(t)⊥¯ µ, (13) N¯t − N¯t which, together with equations (11) and (12), constitute the local conservation laws in QMR. Note that these laws do not depend on source composition, and that even the metric approximation of equation (13) is diﬀerent from its counterpart in metric gravity.

To construct gt from ¯gt we need the 3-vector ﬁeld family vt. Expressed in a GTCS vt by deﬁnition has the components [2, 3]

vj y¯ bj , v =y ¯ h¯ bi bk , (14) (t)≡ t F t (t)ik F F q where v is the norm of vt and bF is a 3-vector ﬁeld found from the equations

a¯k + c−2a¯ a¯k bj a¯j + c−2a¯ a¯j bk 2¯aj =0. (15) F|k Fk F F − F|k Fk F F − F h i h i t0 i ∂ We now define the unit vector field ¯e e¯ i and the corresponding covector field b≡ t b ∂x ¯eb t e¯bdxi along b . Then we have [2, 3] ≡ t0 i F v2 2 g = 1 g¯ , (16) (t)00 − c2 (t)00 2 2 v v t c i ¯ b g(t)0j = 1 2 g¯(t)0j + v (¯ebNi)¯ej , (17) − c t0 1 h − c i

t2 4 v g =g ¯ + c e¯be¯b. (18) (t)ij (t)ij t2 (1 v )2 i j 0 − c

These formulae deﬁne the transformation ¯gt gt. Notice that we have eliminated any → 0 possible t-dependence of N¯t in equations (16)-(18) by setting t = x /c where it occurs. This implies that N does not depend explicitly on t.

6 3 Metrically stationary, axially symmetric systems

In this section we examine the gravitational field interior respectively exterior to an isolated, axially symmetric, spinning source made of perfect fluid. We require that the rotation of the source should have no time dependence apart from the effects coming from the global cosmic expansion. Besides, as for the spherically symmetric, metrically static case treated in [4], we require that the only time dependence of the gravitational field is via the cosmic scale factor. (See the appendix where it is shown that we can neglect any net translatory motion of the source with respect to the cosmic rest frame without loss of generality.) But contrary to the metrically static case, there is a non-zero shift vector field present due to the rotation of the source. However, we require that a GTCS can be 0 found where N¯t and N¯j are independent of x and t. We call this a metrically stationary case. The axial symmetry can be directly imposed on equation (4). Introducing a spherical GTCS x0,ρ,θ,φ where ρ is an isotropic radial coordinate and where the shift vector { } field points in the negative φ-direction, N¯t and N¯φ do not depend on φ and equation (4) takes the form 2 2 2 ¯ ¯ φ ¯ 2 0 2 t ¯ 0 t ¯ 2 dρ 2 2 dst = NφN Nt (dx ) +2 Nφdφdx + 2 Nt ρ2 + ρ dΩ , (19) − t0 t0 1 2 h i h − Ξ0 i where dΩ2 dθ2 + sin2θdφ2, Ξ ct and 0 N¯ ρ < Ξ . The range of ρ is limited for ≡ 0≡ 0 ≤ t 0 both physical and mathematical reasons; truly isolated systems cannot exist in quasi- metric gravity [4]. That is, realistic nontrivial global solutions of the field equations on S3 for isolated systems do not exist, according to the maximum principle applied to closed Riemannian manifolds. However, for some isolated systems, QMR allows exact 3 “semiglobal” solutions on half of S (with the reasonable boundary conditions N¯t(Ξ0) = 1, i and presumably, N¯ (Ξ0) = 0; the same values as for an empty Universe). And although such solutions may be mathematically extended to (almost) the whole of S3, the metric transformation ¯g g is singular at the boundary, so that any transformation based on t→ t the extended solutions becomes mathematically meaningless. The requirement that both 3 ¯gt and gt exist is thus satisfied only for the half of S . Using the definition B¯ N¯ 2, equation (19) may conveniently be rewritten in the form ≡ t 2 2 2 ¯ ¯ 2 2 2 0 2 t ¯ 2 2 0 t dρ 2 2 dst = B (1 V ρ sin θ)(dx ) +2 V ρ sin θdφdx + 2 ρ2 + ρ dΩ , (20) − − t0 t0 1 2 h − Ξ0 i where N¯ V¯ φ . (21) ≡Bρ¯ 2sin2θ

7 Simple expressions for the the non-vanishing components of the extrinsic curvature tensor family may be found from equations (7), (20) and (21). They read (note that the trace

K¯t vanishes)

t 2 2 t 2 2 K¯(t)ρφ = K¯(t)φρ = Bρ¯ sin θV,¯ ρ , K¯(t)θφ = K¯(t)φθ = Bρ¯ sin θV,¯ θ . (22) 2t0 2t0 p p The unknown quantities B¯(ρ, θ) and V¯ (ρ, θ) may now in principle be calculated from the ﬁeld equations and local conservation laws.

3.1 The interior ﬁeld

We will now set up the general field equations interior to the source, which is modelled as a perfect fluid (in general consisting of a mixture of photons and material particles). However, no attempt will be made to find solutions. Note that the metrically stationary condition implies a negligible net rate of energy transfer between photons and material particles. To begin with we consider Tt for a perfect fluid,

T = (˜ρ + c−2p˜)¯u ¯u +˜p¯g , (23) t m t⊗ t t whereρ ˜m is the active mass-energy density in the local rest frame of the ﬂuid andp ˜ is the active pressure. Furthermore ¯u is the 4-velocity vector family in ( , ¯g ) of observers t N t co-moving with the ﬂuid. It is useful to set up the general formula for the split-up of ¯ut into pieces respectively normal to and intrinsic to the FHSs:

2 ⋆ ⋆ w¯ 1 ¯u = γ¯(c¯n + ¯w ), γ¯ (1 )− 2 , (24) t t t ≡ − c2 where ¯wt (with normw ¯) is the 3-velocity family with respect to the FOs. Note that due to the axial symmetry, ¯w points in the φ-direction. Moreover, by deﬁnition the t ± quantity ρm is the passive mass-energy density as measured in the local rest frame of the

ﬂuid and p is the passive pressure. The relationship betweenρ ˜m and ρm is given by

t0 ¯ −1 t Nt ρ˜m, for a fluid of material particles, ρm = 2 (25)  t0 −2 2 N¯ ρ˜ , for the electromagnetic field,  t t m and a similar relationship exists betweenp ˜ and p. The reason why the relationship betweenρ ˜m and ρm is different for a null fluid than for other perfect fluid sources is that gravitational or cosmological spectral shifts of null particles influence their passive mass-energy but not their active mass-energy.

8 The next step is to use equations (23) and (24) to find suitable expressions for the source terms of the field equations (5) and (6). We find

⋆ 2 2 ⋆ 2 i 2 w¯ 2 t0 2 w¯ 2 T ¯ ¯ + Tˆ = γ¯ (1 + )(˜ρ c +p ˜)+2˜p γ¯ (1 + )(¯ρ c +p ¯)+2¯p , (26) (t)⊥⊥ (t)i c2 m ≡t2B¯ c2 m h i whereρ ¯m is the coordinate volume density of active mass andp ¯ is the associated pressure. Moreover we ﬁnd (assuming that the source rotates in the positive φ-direction)

⋆ ⋆ 2 w¯(t)φ 2 t0 2 w¯ 2 T(t)⊥¯ φ = γ¯ (˜ρmc +˜p)= γ¯ ρsinθ (¯ρmc +p ¯). (27) c t√B¯ c The nontrivial parts of the local conservation laws (13) yield

⋆ w¯ B,¯ ⋆ w¯2 p,¯ = ρ¯ c2 3¯p γ¯2V¯ ρsinθ (¯ρ c2 +p ¯) ρ + γ¯2ρ−1 (¯ρ c2 +¯p), ρ − m − − c m 2B¯ c2 m h ⋆ w¯ iB,¯ ⋆ w¯2 p,¯ = ρ¯ c2 3¯p γ¯2V¯ ρsinθ (¯ρ c2 +p ¯) θ + γ¯2cotθ (¯ρ c2 +¯p). (28) θ − m − − c m 2B¯ c2 m h i Note that the metrically stationary condition implies that one must have an equation of state of the form p ρ since otherwiseρ ¯ andp ¯ cannot both be independent of t. ∝ m m We are now in position to set up the field equations (5), (6) for the system in an as simple as possible form. After calculating the necessary derivatives and doing some simple (EM) 1 (EM) 2 algebra we find the two coupled partial differential equations (withp ¯ = 3 ρ¯ c )

2 2 ρ ¯ 1 ¯ 2 3ρ ¯ cotθ ¯ (1 2 )B,ρρ + 2 B,θθ + (1 2 )B,ρ + 2 B,θ − Ξ0 ρ ρ − 2Ξ0 ρ 2 ¯ 2 2 ρ ¯ 2 1 ¯ 2 = B ρ sin θ (1 2 )(V,ρ ) + 2 (V,θ ) − Ξ0 ρ n h i 2 ⋆ w¯2 ⋆ w¯2 + κBρ¯(EM)c2 2γ¯2(1 + )+1 + κS γ¯2(1 + )(¯ρmatc2 +¯pmat)+2¯pmat , (29) 3 m c2 c2 m h i h io ρ2 1 4 5ρ ρ2 B,¯ B,¯ 1 (1 )V,¯ + V,¯ + + (1 ) ρ V,¯ + 3cotθ + θ V,¯ 2 ρρ 2 θθ 2 2 ¯ ρ ¯ 2 θ − Ξ0 ρ ρ − Ξ0 − Ξ0 B B ρ h (EM) i h i 8 ⋆ w¯ ρ¯ c2 ⋆ w¯ (¯ρmatc2 +¯pmat) = κBγ¯2 m +2κSγ¯2 m . (30) 3 c ρsinθ c ρsinθ 2 ¯ 2 Notice that the left hand side of equation (29) is equal to s B, where s is the Laplacian 3 ∇ ∇ compatible with the standard metric on S (with radius equal to Ξ0). To avoid problems with coordinate pathologies along the axis of rotation it would probably be convenient to express equations (28)-(30) in Cartesian coordinates rather than trying to solve them numerically as they stand. However, due to their complexity no attempt to ﬁnd an interior solution will be made in this paper.

9 3.2 The exterior ﬁeld

For illustrative purposes, let us ﬁrst consider exact solutions of equation (29) without source terms for the metrically static, axisymmetric case; we may then set V¯ = 0. If we (ms) (ms) also insist that the solution B¯ (ρ, θ) fulﬁls the boundary condition B¯ (Ξ0, θ)=1 (which should not be taken to be a realistic physical constraint, since true isolated systems do not exist in QMR [4]), it turns out that the solution (on the half of S3) must take the form

2 ¯ 2 ¯(ms) rs0 ρ 2 B (ρ, θ)=1 1 2 1 J2 R2 (3cos θ 1) , (31) − ρ s − Ξ0 − 2ρ − h i where J2 is the (static) quadrupole-moment parameter and the other quantities are as in equation (32) below. It is clear that the source corresponding to this solution is a body which has a nonspherical shape (oblate spheroid) in absence of any rotation. Thus this body cannot be made of perfect fluid, since the source material must be able to support shear forces. From the solution (31) one is in principle able to construct the (ms) counterpart exact “physical” metric family gt using equations (14)-(18). However, the expressions thus obtained are extremely complicated so we will not include them here. A series expansion can be obtained from equation (41) below in the limit of no rotation with a non-zero static quadrupole-moment parameter. Note that the solution (31) can 3 (ms) be extended to (almost) the whole of S but that the associated family gt cannot. Returning to the metrically stationary, axisymmetric case; so far we have not been able to find any exact exterior solution (where V¯ =constant) of equations (29) and (30) 6 (without sources). On the other hand, it is straightforward to find the first few terms of a series solution. Whether or not this series converges to a real solution on the half of S3 is not known. However, it is plausible that such a solution exists in complete analogy with the solution (31). Furthermore, it is not unreasonable to expect that the calculated series expansion converges towards this real solution, since in the limit of no rotation, the first terms of the series are identical to those obtained by writing the solution (31) as a series expansion. The first few terms of the series solution is (for the case where the source spins in the positive φ-direction)

¯2 2 2 ¯ rs0 rs0ρ rs0 2 rs0a0 2 B(ρ, θ)=1 + 2 + J2 R 3 (3cos θ 1) + 4 (3sin θ 1) + , (32) − ρ 2Ξ0 2ρ − 2ρ − ···

r a 3r V¯ (ρ, θ)= s0 0 1+ s0 + , (33) − ρ3 4ρ ··· 10 where J now is the rotationally induced quadrupole-moment parameter and ¯ is the 2 R mean coordinate radius of the source. Furthermore r 2M (EM)GB/c2 +2M matGS/c2 is s0≡ t0 t0 the generalized Schwarzschild radius at epoch t0 deﬁned from the Komar masses [5, 6] (EM) mat Mt0 and Mt0 , i.e.

M mat c−2 N¯ (T mat + Tˆmati) 2N¯ φT mat h¯ d3x, (34) t0 ≡ t0 (t0)⊥¯ ⊥¯ (t0)i − (t0)⊥¯ φ t0 Z Z Z h iq (EM) and a similar formula for Mt0 . Moreover a0 is a length at epoch t0 deﬁned from the (EM) mat angular momentum integrals [6] Jt0 and Jt0 (the integrals are in principle taken over the half of S3 since, as for the solution (31), one expects that any further extension of the series solution would be inconsistent and mathematically irrelevant)

J (EM)GB + J matGS t0 t0 mat −1 φ mat ¯ 3 a0 , Jt0 c ψ T(t0)⊥¯ φ ht0 d x, (35) ≡c(M (EM)GB + M matGS) ≡ t0 t0 Z Z Z q (EM) ¯ ¯ ∂ and a similar formula for Jt0 . Here ht0 is the determinant of ht0 and ψ ∂φ is a Killing ≡ 2 (EM) mat t vector ﬁeld associated with the axial symmetry. Note that Jt = J +J = 2 Jt0 is the t t t0 active angular momentum of the source at epoch t. (The corresponding passive angular mat t mat momentum for a purely material source (i.e., containing no photons) is Lt = t0 Lt0 , where Lmat = c−1 ψφ ¯ mat h¯ d3x, and where ¯ mat is the passive stress-energy t0 T(t0)⊥¯ φ t0 Tt tensor for a purely material source in ( , ¯g ).) We now insert equations (32) and (33) R R R p N t into equation (20). Taking into account a relevant number of terms we get

¯ 2 2 2 2 rs0 rs0ρ rs0 2 rs0a0 2 0 2 dst = 1 + 2 + J2 R 3 (3cos θ 1) 4 cos θ + (dx ) − − ρ 2Ξ0 2ρ − − 2ρ ··· 2 2 t rs0 rs0a0 2 0 t rs0 dρ 2 2 2 (1 + ) sin θdφdx + 2 (1 + ) ρ2 + ρ dΩ . (36) − t0 − 4ρ ··· ρ t0 − ρ ··· 1 2 − Ξ0

To construct gt from ¯gt we need to calculate the vector ﬁeld bF from equation (15) (¯aF and its derivatives may be found from equations (3) and (31)). These calculations get quite complicated so it is convenient to do them by computer. The result is

2 ¯ 2 3 ρ rs0 rs0 3 2 rs0ρ rs0 bF = ρ 1 2 + J2 R2 (3cos θ 1) 2 3 − 2ρ − 4ρ 2ρ − − 4Ξ0 − 8ρ 3r a2 3 ¯ 2r + s0 0 (3sin2θ 2) J R s0 (3cos2θ 1)+ , (37) 2ρ3 − − 2 4ρ3 − ···

3sin(2θ) 3r 3r a2 ρbθ = J ¯ 2(1 s0 + ) s0 0 + . (38) F − ρ 2R − 4ρ ··· − 2ρ ··· 11 Furthermore we need the quantity v deﬁned in equation (14). This may be expressed by

B¯ and its derivatives together with the components of bF . We find c ρ2 ρ2 r c r v = (1 )(B,¯ )2 + ρ−2(B,¯ )2 (1 )−1(bρ )2 +(ρbθ )2 = s0 1+ s0 ¯ 2 ρ θ 2 F F 2B s − Ξ0 s − Ξ0 2ρ 2ρ 2 2 3 ¯ 2 2 rs0 ρ rs0 rs0ρ rs0 2 rs0a0 2 + 2 + 2 + 3 2 J2 R 3 (3cos θ 1) 3 (3sin θ +2)+ . (39) 4ρ 2Ξ0 8ρ − 2Ξ0 − 2ρ − − 2ρ ··· b Finally, to do the transformation shown in equation (18) we need the quantitiese ¯ρ and b b θ e¯θ. Sincee ¯θ is equal to ρbF to the accuracy calculated here, it is sufficient to write down b the expression fore ¯ρ. A straightforward calculation yields 2 2 3 ¯ 2 b rs0 rs0 ρ rs0ρ rs0 rs0 2 e¯ρ =1 2 + 2 + 2 3 + J2 R 3 (3cos θ 1) + , (40) − 2ρ − 8ρ 2Ξ0 4Ξ0 − 16ρ 4ρ − ··· and the transformations (16)-(18) then yield, to desired accuracy 2 ¯2 2 2 2 rs0 rs0 rs0ρ rs0 2 rs0a0 2 dst = 1 2 + 2 + J2 R 3 (3cos θ 1) 4 cos θ − − ρ − 2ρ 2Ξ0 2ρ − − 2ρ 4 2 3rs0 rs0 0 2 t rs0 rs0a0 2 0 + 4 2 + (dx ) 2 (1 + ) sin θdφdx 16ρ − 2Ξ0 ··· − t0 − 4ρ ··· ρ 2 2 2 3 t rs0 rs0 dρ rs0 rs0 2 2 + 2 (1 + + 2 + ) ρ2 + (1 + O( 3 ))ρ dΩ . (41) t0 ρ ρ ··· 1 2 − ρ ρ − Ξ0 3 rs0 Note that to O( ρ3 ) or higher, the spatial metric family ht is not diagonal in these coordinates. It may be convenient to express the metric family (41) in an “almost Schwarzschild” radial coordinate r defined by

r ρ2 r r2 r3 r ρ r ρ 1 s0 1 = 1 s0 s0 s0 + s0 + ρ, v 2 2 3 2 ≡ u − ρ s − Ξ0 − 2ρ − 8ρ − 16ρ 4Ξ0 ··· u t 2 rs0 rs0 rs0r ρ = 1+ + 2 2 + r. (42) 2r 8r − 4Ξ0 ··· That is, to the order in small quantities considered in equation (41), at epoch t0 the surface area of spheres centered on the origin is equal to 4πr2. Expressed in the new radial coordinate (41) reads 3 ¯ 2 2 2 2 rs0 3rs0 rs0r rs0 2 3rs0 rs0a0 2 dst = 1 + 3 + 2 + J2 R 3 (3cos θ 1)(1 ) 4 cos θ − − r 8r 2Ξ0 2r − − 2r − 2r 4 2 rs0 rs0 0 2 t 3rs0 rs0a0 2 0 4 2 + (dx ) 2 (1 + ) sin θdφdx −16r − 2Ξ0 ··· − t0 − 4r ··· r 2 2 2 3 t rs0 rs0 r 2 2 rs0 2 + 2 (1 + + 2 + 2 + )dr + r (1 + O( 3 ))dΩ . (43) t0 r 4r Ξ0 ··· r 12 This expression represents the gravitational field outside an isolated, metrically stationary, axially symmetric spinning source made of perfect fluid (obeying an equation of state of the form p ρ ) to the given accuracy in small quantities. Note in particular the ∝ m presence of a tidal term containing the free parameter J2 describing the effect of source deformation due to the rotation. This tidal term has a counterpart in Newtonian gravitation. (However, to higher order there is also a term due to the rotation itself, and this term has no counterpart in Newtonian gravitation.) The existence of a tidal term means that the metric family (43) has built into itself the necessary flexibility to represent the gravitational field exterior to a variety of sources. That is, since the exact equation of state describing the source is not specified, the effect of the rotation on the source and thus its quadrupole-moment should not be exactly known either, since this effect depends on material properties of the source. On the other hand, which is well-known, no such flexibility is present in the Kerr metric, meaning that the Kerr metric can only represent the gravitational field outside a source which material properties are of no concern; e.g., a spinning black hole.

4 The geodetic and Lense-Thirring eﬀects

To calculate the predicted geodetic and Lense-Thirring effects within the quasi-metric framework we can use the metric family (43) with some extra simplifications. We thus examine the behaviour of a small gyroscope in orbit around a metrically stationary, axially symmetric, isolated source. We may assume that the source is so small that any dependence on the global curvature of space can be neglected. Furthermore we assume that the exterior gravitational field of the source is so weak that it can be adequately represented by equation (43) with the highest order terms cut out, i.e.,

r ¯ 2r ds 2 = 1 s0 + J R s0 (3cos2θ 1) + (dx0)2 t − − r 2 2r3 − ··· 2 t rs0a0 2 0 t rs0 2 2 2 2 sin θdφdx + 2 (1 + + )dr + r dΩ . (44) − t0 r t0 r ··· In this section we calculate the predicted geodetic effect. In this case equation (44) can be simplified even further by assuming that the gravitational field is spherically symmetric, i.e., that the spin of the source can be neglected. We then set J2 = a0 = 0 in equation (44) and it takes the form

2 2 0 2 t 2 2 2 dst = B(r)(dx ) + 2 A(r)dr + r dΩ , (45) − t0 13 where A(r) and B(r) are given as series expansions from equation (44) (with spin parameters neglected). Our derivation of the geodetic eﬀect in QMR will be a counterpart to a similar calculation valid for GR and presented in [7]. To simplify calculations we assume that the gyroscope orbits in the equatorial plane and that the orbit is a circle with constant radial coordinate r = R. Furthermore the gyroscope has spin St and 4-velocity ut. Then the norm S √S S is constant along its world line and moreover normal to u , i.e. ∗≡ t· t t dxi S u =0, S = S . (46) t· t ⇒ (t)0 − (t)i dx0

The equation of motion for the spin St is the equation of parallel transport along its world line in quasi-metric space-time, i.e. µ ⋆ dS ⋆ ⋆ (t) µ λ ν µ λ dt ut St =0, = ΓλνS(t)u(t) ΓλtS(t) . (47) ∇ ⇒ dτt − − dτt

dφ dφ Next we deﬁne the angular velocity of the gyroscope. This is given by Ωt dt = c dx0 . 0 ≡ Thus uφ dφ = c−1Ω dx . Now a constant of motion J for the orbit is given by the (t)≡ dτt t dτt equation [3, 4] (using the notation ′ ∂ ) ≡ ∂r ′ 3 t 2 B (R)R R Ωt = B(R)Jc, J = 2 , (48) t0 2B (R) where the last expression is shown in [8]. Equations (48) then yield

′ t0 B (R) Ωt = c. (49) t r 2R Also, from the fact that u u = c2, we ﬁnd t· t − 0 0 dx c u(t) = = , (50) dτt B(R) 1 B′(R)R − 2 q and using (50), equation (46) yields

0 t −1 1 ′ 3 φ S(t) = B (R) B (R)R S(t). (51) t0 r2 We now insert the expressions found above into equation (47). (The relevant connection coefficients can be found in [3] or [4].) Equation (47) then yields a set of 2 coupled, first order ordinary differential equations of the form

d t r φ d t φ r S(t) = f(R)S(t), S(t) = g(R)S(t), (52) dt t0 dt t0 − h i h i 14 where the functions f(R), g(R) are given by

′ c 1 ′ −1 1 ′3 3 B (R) f(R) B (R)R B (R) B (R)R , g(R) 3 c. (53) ≡A(R) 2 − 8 ≡r 2R r r A solution of the system (52) can be found by computer. Assuming that St points in the

(positive) radial direction at epoch t0, the solution of (52) reads

r t0 −1/2 t S(t) = S∗A (R)cos ωSt0ln( ) , (54) t t0 h i

φ t0 S∗Ωt0 t S(t) = sin ωSt0ln( ) , (55) − t A(R)RωS t0 h i where p

′ −1/2 B (R)R ωS f(R)g(R)= A (R) 1 Ωt0 . (56) ≡ s − 2B(R) p 2π After one complete orbit of the gyroscope t = t0 + Ωt , and the angle between St and a unit vector er in the radial direction is given by

St 2π α = arccos( er)= ωSt0ln[1 + ]. (57) S∗ · Ωtt0 The diﬀerence ∆φ between a complete circle of 2π radians and the angular advancement 2MGN of S will then be (with r 2 , where G is Newton’s constant) t s0≈ c N

2 3 3rs0 πR 2R 3πMt0 GN 2π R ∆φ =2π α =2π + 2 + . (58) − 4R − Ξ0 rs0 ··· ≈ c R − t0 sMt0 GN ··· h r i We see that there is a quasi-metric correction term in addition to the usual GR result. Unfortunately, the difference amounts only to about 5 10−5′′ per year for a satellite − × orbiting the Earth, i.e. the predicted correction is too small by a factor about ten to be detectable by Gravity Probe B. One may also calculate the Lense-Thirring effect for a gyroscope in polar orbit, using equation (44). But except from the variable scale factor, the off-diagonal term in (44) is the same as for the Kerr metric. Any correction term to the Lense-Thirring effect should therefore depend on the inverse age of the Universe and thus be far too small to be detectable. But notice that, for a gyroscope in orbit around the Earth, there is also an extra contribution term of the type shown in equation (58), to the geodetic effect coming from the Earth’s orbit around the Sun. Numerically this correction term is similar to the correction term found above.

15 5 Conclusion

In GR, very many exterior and interior solutions of Einstein’s equations are possible in principle for axisymmetric stationary systems. The problem is to find physically reasonable solutions where the exterior and interior solutions join smoothly to form an asymptotically flat, global solution. Such solutions would be candidates for modelling isolated spinning stars. And although no exact solution having the desired properties has been found so far, accurate analytical approximations exist, and also numerical solutions of the full Einstein equations, see, e.g., [9]. The quasi-metric counterpart to axially symmetric, stationary systems in GR, is metrically stationary, axially symmetric systems. This research subject is largely unexplored. However, this paper contains some basic results for such systems. That is, we have set up the relevant equations for a metrically stationary, axially symmetric isolated source within the quasi-metric framework, both interior and exterior to the source. The first few terms of what may turn out to be a series solution was found for the exterior part (however, it is not known if such a solution exists or whether the calculated series converges towards it). The biggest difference between the candidate series solution and the

Kerr metric is the presence of a term containing the free parameter J2 representing the quadrupole-moment of the source. Such a free parameter is necessary to ensure sufficient flexibility so that the solution does not unduly constrain the nature of the source. On the other hand, the multipole moments of the Kerr metric are fixed. This means that, unlike the Kerr metric, the candidate metric family found in this paper may represent the gravitational field exterior to a variety of sources which equation of state satisfies p ρ . ∝ m Besides, in the limit where the rotation of the source vanishes, the metrically stationary, axially symmetric candidate solution becomes identical to the spherically symmetric, metrically static exterior solution found in [4] (to the given accuracy). This means that the source’s quadrupole-moment vanishes in the limit of no rotation and that the source should be unable to support shear forces. Thus for such a source, its quadrupole-moment is purely due to rotational deformation. It is also possible that part of the source’s quadrupole-moment is static, in which case the source cannot be made of perfect fluid. One possible limit of no rotation is then given from equation (31). Since the field equations (5), (6) in some sense represent only a (generalized) subset of the full Einstein equations [2, 3], one would expect that the number of possible solutions of equations (29) and (30) (with and without sources) are considerably smaller than for the GR counterpart. In particular, one would expect that the number of unphysical solutions are much smaller, and since the metric family ¯gt is constrained to take the form

16 (4), that the problem of smooth matching between interior and exterior solutions would be more or less absent. However, whether this is correct or not will only be known from further research.

References [1] N. Stergioulas, Living Reviews in Relativity 6, 3 (2003). [2] D. Østvang, Grav. & Cosmol. 11, 205 (2005) (gr-qc/0112025). [3] D. Østvang, Doctoral Thesis, (2001) (gr-qc/0111110). [4] D. Østvang, Grav. & Cosmol. 13, 1 (2007) (gr-qc/0201097). [5] A. Komar, Phys. Rev. 113, 934 (1959). [6] R.M. Wald, General Relativity, The University of Chicago Press (1984). [7] J.B. Hartle, Gravity: an introduction to Einstein’s general relativity, Addison-Wesley (2003). [8] S. Weinberg, Gravitation and Cosmology, John Wiley & Sons Inc. (1972). [9] E. Berti, F. White, A. Manipoulou, M. Bruni, MNRAS 358, 923 (2005).

A Preferred-frame eﬀects are absent in QMR

Both in this paper and in [4] it is assumed that the gravitating source has no net translatory motion with respect to the FOs. However, a given object will usually have a non-zero velocity with respect to the cosmic rest frame (i.e., a non-zero dipole moment in the cosmic microwave background will in general be observed for observers being at rest with respect to the object). Since the FOs do not move on average with respect to the cosmic rest frame it might be natural to assume that an isolated, gravitating source could have a net translatory motion with respect to the FOs. This would represent a “preferred-frame” effect which might possibly be measurable. However, as we shall see, an isolated source can never have a net translatory motion with respect to the FOs in QMR, so there will be no preferred-frame effects. The reason for this is that, while the global time function t and the FHSs are quantities given from the overall structure of quasi-metric space-time, the motion of the FOs is determined by the requirement that they always move orthogonally to the FHSs. This requirement is solely determined from the matter distribution via the relevant equations yielding gt; there are no other factors involved. We now illustrate this with a specific calculation. Let us consider an isolated, metrically stationary, gravitating source moving with constant speed U¯ with respect to a GTCS where the cosmic substratum is at rest on average.

17 That is, the cosmic rest frame may be represented by a cylindrical GTCS (x0,ξ′, z′,φ′) oriented such that the source moves in the positive z′-direction with coordinate velocity ′ dz t0 U¯ dx0 = t c , where we for simplicity have neglected the global curvature of the FHSs. This system has axial symmetry and the shift vector ﬁeld has a z′-component only, so from equation (24) we ﬁnd

′ ¯ z′ c dz t0 ¯ z′ ¯ z′ U w¯ w¯(t) = ( 0 + N ), N = − . (A.1) N¯t dx t ⇒ − c The line element family (4) then takes the form (B¯ N¯ 2) ≡ t ¯ 2 ¯ 2 2 ¯ (U w¯) 0 2 t U w¯ ′ 0 t ′2 ′2 ′2 ′2 dst = B (1 −2 )(dx ) 2 − dz dx + 2 dξ + dz + ξ dφ .(A.2) − − c − t0 c t0 h i Here B¯ andw ¯ are functions of x0, ξ′ and z′ but not of φ′. We now assume that the stationary nature of the system makes it possible to eliminate the dependence on x0 by making a coordinate transformation z′ z determined from the differentials → t U¯ dz dz′ t U¯ dz = dz′ 0 dx0, = 0 =0, (A.3) − t c ⇒ dx0 dx0 − t c confirming that the source has no net velocity with respect to the new GTCS. Integrating equation (A.3) we find the new coordinates

0 ¯ x 0′ 0 ′ ′ U dx ′ ¯ x ′ ξ = ξ , z = z t0 ′ = z Ut0ln 0 , φ = φ , (A.4) − c 0 t(x0 ) − x Zx0 0 0 where x0 = ct0. Using equation (A.3) we now write the metric family (A.2) on the form

2 2 2 ¯ w¯ 0 2 t w¯ 0 t 2 2 2 2 dst = B (1 2 )(dx ) +2 dzdx + 2 dξ + dz + ξ dφ , (A.5) − − c t0 c t0 h i where B¯ andw ¯ are functions of ξ and z only. Note that the performed coordinate transformation yields a new shift vector ﬁeld pointing in the z-direction with magnitude w¯ c . From equations (7), (8) and (A.5) we may calculate the non-vanishing components of the extrinsic curvature tensor. We ﬁnd

t w,¯ z w¯ B,¯ z t w,¯ ξ K¯(t)zz = B¯ + , K¯(t)ξz = K¯(t)zξ = B¯ , (A.6) t0 c 2c √B¯ 2t0 c p p

2 t w¯ B,¯ z t w¯ ξ B,¯ z K¯(t)ξξ = , K¯(t)φφ = , (A.7) 2t0 c √B¯ 2t0 c √B¯

18 t w,¯ 3¯w B,¯ K¯ = 0 B¯−1/2 z + z . (A.8) t t c 2c B¯ Using equations (A.6), (A.7) and (A.8) we may now compute the necessary quantities entering into the field equations (5), (6). After some calculations the field equations yield B,¯ w¯2 B,¯ 3¯w2 B,¯ 2 B,¯ ξξ + (1 3 ) zz + z + ξ B¯ − c2 B¯ 2c2 B¯ ξB¯ w¯w,¯ w,¯ 2 w,¯ 2 B,¯ w¯w,¯ 2 zz 2 z 2 ξ 4 z z =0, (A.9) − c2 − c − c − B¯ c2 1 3 B,¯ B,¯ 2B,¯ B,¯ 2 w,¯ + + ξ w,¯ z w,¯ zz 3 z w¯ =0, (A.10) ξξ ξ 2 B¯ ξ − B¯ z − B¯ − B¯ h i h i B,¯ 2B,¯ B,¯ B,¯ w,¯ + z w,¯ + ξz 3 ξ z w¯ =0. (A.11) ξz B¯ ξ B¯ − B¯2 Equations (A.9)-(A.11) are far too complicatedh to havei any hope of finding an exact solution. But what we can do is to look for a weak field solution. That is, for weak field we can set (since by hypothesis, the FOs are at rest with respect to the cosmic rest frame far away from the object) B¯ 1, w¯ = U¯ ǫ,¯ ǫ¯ 1, (A.12) ≈ − | |≪ and neglect all non-linear terms in equations (A.9)-(A.11). Using equation (A.12) the weak field versions of equations (A.9)-(A.11) then read U¯ 2 1 U¯ B,¯ +(1 3 )B,¯ + B,¯ +2 ǫ,¯ =0, (A.13) ξξ − c2 zz ξ ξ c2 zz 1 1 ǫ,¯ +2B,¯ + ¯ǫ, =0, ¯ǫ, =2U¯B,¯ . (A.14) U¯ ξξ zz Uξ¯ ξ ξz ξz Sinceǫ ¯ is required to vanish far from the source, (A.14) yields 1 ǫ¯ =2U¯(B¯ 1), B,¯ +B,¯ + B,¯ =0, (A.15) − ξξ zz ξ ξ but this expression forǫ ¯ inserted into equation (A.13) yields U¯ 2 1 B,¯ +(1 + )B,¯ + B,¯ =0, (A.16) ξξ c2 zz ξ ξ which is inconsistent unless U¯ = 0. Equations (A.9)-(A.11) thus have no solution unless w¯ = 0 in which case the solution is equivalent to that of the metrically static, axially symmetric case. That is, the gravitational field of an isolated source does not depend on the source’s net motion with respect to the cosmic rest frame; such motions may be neglected without loss of generality.